The proof of this theorem will be provided in Appendix B. This theorem gives the multiplicity of the root zero in any \(Q_{N}^{[m]}(z)\)  polynomial. It also shows that the root structure of \(Q_{N}^{[m]}(z)\)  is invariant under \(2\pi /(2m+1)\) -angle rotation in the complex \(z\)  plane. In the particular case of the original Yablonskii-Vorob'ev polynomials \(Q_{N}(z)\)  where \(m=1\) , the above theorem shows that \(0\le N_0\le 1\) . This means that zero is either not a root or a simple root for \(Q_{N}(z)\) , in agreement with previous results in [1]}, [2]}.
